In fuzzy set theory, an element may belong to a set only partially,
in some degree. We show that the partial membership can be modelled
as a subset of a special abstract universe so that the membership
degree can be defined as a measure on that universe. As a result of
that, the basic set operations are uniquely established since there
is no freedom in choosing a t-norm for intersection, a t-conorm for
union etc. Moreover, a relationship between any pair of fuzzy sets
becomes visible that determines uniquely the function of a set
operation on membership degrees. It is shown that the membership
degrees of an intersection are governed by copulas and the union
operation is ruled by copula dualities. Contrary to Zadeh’s
definition of fuzzy sets, our operations on fuzzy sets form a
Boolean algebra.